Re: Logarithm of transfinite numbers

Virgil wrote:
In article <MPG.1e9df60b9a559eac98abf0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

MoeBlee said:
Tony Orlow wrote:
Actually, I can axiomatically state these things and treat Big'un
as a primitive, and if I can derive no contradictions, and can
derive useful results, you have nothing to complain about.

That would be good. But since you criticize other mathematics on
the basis that its axioms aren't true as statements about a
fundamental reality, then your own axioms are subject to such
scrutiny too. I don't know why you would think it is so manifestly
true that there is an object that exists as a fundamental reality
that is the length of the real line but (if you do hold this:) that
there isn't an object that is the set of counting numbers.

Yes, that's not a vacuous point, and one I can appreciate. Assuming a
length to the real number line when it has no discernible ends does
seem somewhat arbitrary, and that's why it needs to be assumed a
priori, since it's not really a derivable value.

Thus TO is claiming that it is legitimate for HIM to assume things
arbitrarily but not for anyone else?


But, if we say there
exists this infinite line, then there is SOME length to it

Why does something that does not have ends have to have something that
requires that it have ends?



It appears that the assumption that this
value is not only the length of the line, but also the number of
points within any unit segment of it

It does not appear so to us, so that we will require formal proof of
that claim.





Oh. Define 0.

The unique x(Ay ~yex).

That sounds like a definition of the null set ala von Neumann, but
not necessarily going to the heart of what 0 is. Really, I was
referring to its use in the Peano axioms, where is is taken as a
primitive. I see no reason why it can't be taken as an assumed
primitive and a starting place, and actually see this as a natural
starting place for mathemtics as a whole. Start with nothing.

That is precisely what von Neumann did, and is precisely what TO has
violently objected to in the past only because it showed how stupid one
of TO's arguments was.

What's your point?

We don't need protection from Tony here, Virgil, the above being one of
your less abusive posts. I think Tony is sincere and quite pleasant,
knowledgeable and sincere.

I suppose you won't reply. Do you have a point? It's probable that
you should assume, say, a basically high level of mathematical
sophistication of your readers, were you rational, perhaps in
combination with social.

Don't you have anything positive to say? That does not reflect well,
because of the negative things, where saying nothing is not necessarily
the same thing as saying nothing nice.

There's a good point in saying that there are as many reals, on the
line, between zero and one as there are naturals. Then when you sum
their values over the naturals as is the generally understood method in
the integral calculus or "Cholera bacilli analysis", that gives perfect
and generally perfectly expected results. That's where the integral
bar is an S for summation.

Basically post-Cauchy-Weierstrass in the quest for foundations of the
eighteenth and nineteenth centuries certain results and counterexamples
led to basically denial of the quite real utility, and thus in a way
quite obvious formal soundness, of the infinitesimal analysis. That is
where at the same time its justification is disguised as
delta-epsilonics, with basically the exact same notion and quite
widespread usage of the Leibniz notation, how in the limit via infinite
induction it works, including ratios of infinitesimals, eg dx/dy.
Consider L'Hospital(e), and the quadrature (squaring, in 2-D), for
different reasons, and the same.

Obviously in my theory the points are polydimensional, with being one-
or two-sided on the line, in continuity one-sided and with weight over
the summation index of the index' reciprocal, scalar, in the
polydimensional: inverse, with geometry in the large and small, the
very.

There's something new, for you.

The universe is infinite, infinite sets are equivalent. Familiar? How
about the notion that there is a universe, would you agree that there
is a universe. The more comprehensively it's examined the larger it
gets. The more closely subatomic particles are examined the smaller
they get. (Time stops for particle/waves with the HUP. Please explain
gauge invariance.) Right: start with nothing, don't care. That's so
it goes.

Obviously in my theory of axiomless natural deduction, currently A
theory, there's only one theory with no axioms. Conveniently, only and
all true statements are theorems.

Ciao,

Ross F.

.

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